# How to Find a Hypotenuse

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The hypotenuse is the side that is opposite of the right angle or the longest side of the right triangle. The other two sides are called the opposite and adjacent sides. This is a step-by-step guide that will teach you how to find a hypotenuse when you know the other two sides of the right triangle and also when you know an angle and side.

Right Triangle: If a triangle contains one angle that is exactly 90 degrees then it is a right triangle.

Trigonometric Ratios:
Sine: sin(θ) = Opposite / Hypotenuse
Cosine: cos(θ) = Adjacent / Hypotenuse
Tangent: tan(θ) = Opposite / Adjacent

Sides of an Angle:
– Adjacent is next to the angle.
– Opposite is opposite to the angle.
– The longest side is the hypotenuse

## How to Find Hypotenuse when we know the Sides

1. Make sure the triangle is a right triangle.

2. Recall the Pythagorean Theorem. (a2 + b2 =c2)

3. Look at the triangle and identify the values for a, b and c.

4. Plugin the values of legs(a&b) into the Pythagorean Theorem.

5. Solve the equation to get the value of c.

For Example,

$\\\mathbf{a^{2}&space;+&space;b^{2}&space;=&space;c^{2}}&space;\\&space;\\\mathbf{5^{2}&space;+&space;12^{2}&space;=&space;c^{2}}&space;\\&space;\\\mathbf{25&space;+&space;144&space;=&space;c^{2}}&space;\\&space;\\\mathbf{169&space;=&space;c^{2}}&space;\\&space;\\\mathbf{\sqrt{169}&space;=&space;\sqrt{c^{2}}}&space;\\&space;\\\mathbf{\13&space;=&space;c}$

## How to Find Hypotenuse when we know a Side and an Angle

1. Understand the sides of an angle.

2. Recall the trigonometric ratios.

3. Apply the Sine trigonometric relationship to find the hypotenuse.

4. Solve the equation to get the value of Hypotenuse.

For Example,

$\\\mathbf{sin&space;\70\degree&space;=&space;\frac{15}{c}}&space;\\&space;\\\mathbf{c&space;\&space;\&space;=&space;\&space;\frac{15}{sin&space;\70\degree}}&space;\\&space;\\\mathbf{c&space;\&space;\&space;=&space;\&space;\frac{15}{0.7738}}&space;\\&space;\\\mathbf{c&space;\&space;\&space;=&space;\&space;\frac{15}{0.7738}}&space;\\&space;\\\mathbf{c&space;\&space;\&space;=&space;\&space;\19.38}$